Attitude determination finds the rotation between the coordinate system of a known object and that of a sensed portion of its surface. Orientation-based representations record 3-D surface properties as a function of position on the unit sphere. They are useful in attitude determination because they rotate in the same way as the object rotates. Three such representations are defined using, respectively, the support function and the first and second curvature functions. The curvature representations are unique for smooth, strictly convex objects. The support function representation is unique for any convex object.
The essential mathematical basis for these representations is provided. The paper extends previous results on convex polyhedra to the domain of smooth, strictly convex surfaces. Combinations of the support function of a known object with curvature measurements from a visible surface transform attitude determination into an optimization problem for which standard numerical solutions exist.
Dense measurements of surface curvature are required. Surface data can be obtained from laser range finding or from shape-from-shading methods, including photometric stereo. A proof-of-concept system has been implemented and experiments conducted on a real object using surface orientation and surface curvature data obtained directly from photometric stereo.
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